scholarly journals Querying a Matrix through Matrix-Vector Products

2021 ◽  
Vol 17 (4) ◽  
pp. 1-19
Author(s):  
Xiaoming Sun ◽  
David P. Woodruff ◽  
Guang Yang ◽  
Jialin Zhang
Keyword(s):  
Linear Algebra ◽  
Incidence Matrix ◽  
Communication Complexity ◽  
Property Testing ◽  
Distributed Computation ◽  
Maximum Eigenvalue ◽  
Graph Problems ◽  
The Matrix ◽  
The Right ◽  
Matrix Vector

We consider algorithms with access to an unknown matrix M ε F n×d via matrix-vector products , namely, the algorithm chooses vectors v 1 , ⃛ , v q , and observes Mv 1 , ⃛ , Mv q . Here the v i can be randomized as well as chosen adaptively as a function of Mv 1 , ⃛ , Mv i-1 . Motivated by applications of sketching in distributed computation, linear algebra, and streaming models, as well as connections to areas such as communication complexity and property testing, we initiate the study of the number q of queries needed to solve various fundamental problems. We study problems in three broad categories, including linear algebra, statistics problems, and graph problems. For example, we consider the number of queries required to approximate the rank, trace, maximum eigenvalue, and norms of a matrix M; to compute the AND/OR/Parity of each column or row of M, to decide whether there are identical columns or rows in M or whether M is symmetric, diagonal, or unitary; or to compute whether a graph defined by M is connected or triangle-free. We also show separations for algorithms that are allowed to obtain matrix-vector products only by querying vectors on the right, versus algorithms that can query vectors on both the left and the right. We also show separations depending on the underlying field the matrix-vector product occurs in. For graph problems, we show separations depending on the form of the matrix (bipartite adjacency versus signed edge-vertex incidence matrix) to represent the graph. Surprisingly, very few works discuss this fundamental model, and we believe a thorough investigation of problems in this model would be beneficial to a number of different application areas.

scholarly journals An Inverse Matrix Formula in the Right-Quantum Algebra

10.37236/747 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Matjaž Konvalinka
Keyword(s):  
Linear Algebra ◽  
Quantum Algebra ◽  
Matrix Inverse ◽  
Bijective Proof ◽  
The Matrix ◽  
Matrix Formula ◽  
Fundamental Transformation ◽  
The Right ◽  
Quantum Matrix ◽  
Master Theorem

The right-quantum algebra was introduced recently by Garoufalidis, Lê and Zeilberger in their quantum generalization of the MacMahon master theorem. A bijective proof of this identity due to Konvalinka and Pak, and also the recent proof of the right-quantum Sylvester's determinant identity, make heavy use of a bijection related to the first fundamental transformation on words introduced by Foata. This paper makes explicit the connection between this transformation and right-quantum linear algebra identities; we give a new, bijective proof of the right-quantum matrix inverse theorem, we show that similar techniques prove the right-quantum Jacobi ratio theorem, and we use the matrix inverse formula to find a generalization of the (right-quantum) MacMahon master theorem.

Selecting optimal SpMV realizations for GPUs via machine learning

2021 ◽  
pp. 109434202199073
Author(s):  
Ernesto Dufrechou ◽  
Pablo Ezzatti ◽  
Enrique S Quintana-Ortí
Keyword(s):  
Machine Learning ◽  
Sparse Matrix ◽  
Machine Learning Techniques ◽  
Optimal Method ◽  
Learning Techniques ◽  
General Rules ◽  
Machine Learning Approach ◽  
The Matrix ◽  
Time And Energy ◽  
Matrix Vector

More than 10 years of research related to the development of efficient GPU routines for the sparse matrix-vector product (SpMV) have led to several realizations, each with its own strengths and weaknesses. In this work, we review some of the most relevant efforts on the subject, evaluate a few prominent routines that are publicly available using more than 3000 matrices from different applications, and apply machine learning techniques to anticipate which SpMV realization will perform best for each sparse matrix on a given parallel platform. Our numerical experiments confirm the methods offer such varied behaviors depending on the matrix structure that the identification of general rules to select the optimal method for a given matrix becomes extremely difficult, though some useful strategies (heuristics) can be defined. Using a machine learning approach, we show that it is possible to obtain unexpensive classifiers that predict the best method for a given sparse matrix with over 80% accuracy, demonstrating that this approach can deliver important reductions in both execution time and energy consumption.

MRHS solver based on linear algebra and exhaustive search

2018 ◽  
Vol 12 (3) ◽  
pp. 143-157 ◽  
Author(s):  
Håvard Raddum ◽  
Pavol Zajac
Keyword(s):  
Linear Algebra ◽  
Matrix Representation ◽  
Equation System ◽  
Exhaustive Search ◽  
Binary Matrix ◽  
Algebraic Attacks ◽  
The Matrix ◽  
Speed Up ◽  
Symmetric Key

Abstract We show how to build a binary matrix from the MRHS representation of a symmetric-key cipher. The matrix contains the cipher represented as an equation system and can be used to assess a cipher’s resistance against algebraic attacks. We give an algorithm for solving the system and compute its complexity. The complexity is normally close to exhaustive search on the variables representing the user-selected key. Finally, we show that for some variants of LowMC, the joined MRHS matrix representation can be used to speed up regular encryption in addition to exhaustive key search.

DMRG Approach to Fast Linear Algebra in the TT-Format

2011 ◽  
Vol 11 (3) ◽  
pp. 382-393 ◽  
Author(s):  
Ivan Oseledets
Keyword(s):  
Linear Systems ◽  
Linear Algebra ◽  
Direct Method ◽  
Fast Algorithms ◽  
Approximate Solutions ◽  
Vector Product ◽  
The Matrix ◽  
Minimization Scheme

AbstractIn this paper, the concept of the DMRG minimization scheme is extended to several important operations in the TT-format, like the matrix-by-vector product and the conversion from the canonical format to the TT-format. Fast algorithms are implemented and a stabilization scheme based on randomization is proposed. The comparison with the direct method is performed on a sequence of matrices and vectors coming as approximate solutions of linear systems in the TT-format. A generated example is provided to show that randomization is really needed in some cases. The matrices and vectors used are available from the author or at http://spring.inm.ras.ru/osel

scholarly journals Improvement of the Impact Properties of Composite Laminates by Means of Nano-Modification of the Matrix—A Review

2018 ◽  
Vol 8 (12) ◽  
pp. 2406 ◽  
Author(s):  
Hamed Saghafi ◽  
Mohamad Fotouhi ◽  
Giangiacomo Minak
Keyword(s):  
Composite Laminates ◽  
Cost Effective ◽  
Careful Consideration ◽  
Impact Performance ◽  
Compression After Impact ◽  
The Matrix ◽  
The Right ◽  
2D And 3D ◽  
The Impact ◽  
Selection Of

This paper reviews recent works on the application of nanofibers and nanoparticle reinforcements to enhance the interlaminar fracture toughness, to reduce the impact induced damage and to improve the compression after impact performance of fiber reinforced composites with brittle thermosetting resins. The nanofibers have been mainly used as mats embedded between plies of laminated composites, whereas the nanoparticles have been used in 0D, 1D, 2D, and 3D dimensional patterns to reinforce the matrix and consequently the composite. The reinforcement mechanisms are presented, and a comparison is done between the different papers in the literature. This review shows that in order to have an efficient reinforcement effect, careful consideration is required in the manufacturing, materials selection and reinforcement content and percentage. The selection of the right parameters can provide a tough and impact resistant composite with cost effective reinforcements.

The inflammatory macrophage: a story of Jekyll and Hyde

2002 ◽  
Vol 104 (1) ◽  
pp. 27-38 ◽  
Author(s):  
Jeremy S. DUFFIELD
Keyword(s):  
Apoptotic Cells ◽  
Tissue Healing ◽  
Immune Mediated ◽  
Selective Depletion ◽  
The Matrix ◽  
Matrix Components ◽  
The Right ◽  
Extracellular Structures ◽  
Two Populations ◽  
Inflammatory Macrophage

Recent investigations have highlighted new roles for the macrophage (Mϕ) in the biology of inflammation. Selective depletion of Mϕs from inflamed sites has confirmed their predominant role in immune-mediated damage. The components of this injury have been dissected. Mϕs mediate death of stromal, parenchymal and other immune cells by engaging the death programme, resulting in apoptosis. In addition, Mϕs induce destruction of matrix and extracellular structures both directly and indirectly by inducing stromal cells to release matrix metalloproteinases. However, there is another side to the inflammatory Mϕ. Evidence is provided that Mϕs at the same sites possess the ability to aid cell proliferation, secrete and stabilize new matrix components and induce resident cells to secrete matrix components themselves. Mϕ phagocytosis of apoptotic cells brings about a change from the cell-killing matrix-degrading cell to the matrix-generating cell-proliferating tissue-healing cell. Just as both Mϕ types are necessary at the inflamed site, the right balance of these two populations is required for healing and resolution. Evidence of excessive inflammation as a manifestation of impaired phagocytosis of apoptotic cells emphasizes that defects in the transition from one Mϕ type to another may account for the uncontrolled excessive inflammation seen in disease. Recent insights into the mechanisms by which apoptotic cells signal the change of function to the Mϕ offer the prospect of novel targets for manipulation of Mϕs in the inflamed tissue.

Application of Spline Collocation Method in Analysis of Beam and Continuous Beam

2003 ◽  
Vol 19 (2) ◽  
pp. 319-326 ◽  
Author(s):  
Lai-Yun Wu ◽  
Yang-Tzung Chen
Keyword(s):  
Collocation Method ◽  
Differential Quadrature Method ◽  
Spline Functions ◽  
Continuous Beam ◽  
Spline Collocation ◽  
Spline Collocation Method ◽  
The Matrix ◽  
Weighting Coefficients ◽  
Matrix Vector ◽  
Excellent Accuracy

ABSTRACTIn this paper, spline collocation method (SCM) is successfully extended to solve the generalized problems of beam structures. The spline functions in SCM are re-formulated by finite difference method in a systematical way that is easily understood by engineers. The manipulation of SCM is further simplified by the introduction of quintic table so that the matrix-vector governing equation can be easily formulated to solve the weighting coefficients. SCM is first examined by the problems of a generalized single-span beam undergoing various types of loadings and boundary conditions, and it is then extended to the problems of continuous beam with multiple spans. By comparing with available analytical results, differential quadrature method (DQM), if any, excellent accuracy in deflection is achieved.

Advanced three-dimensional electromagnetic modeling using a nested integral equation approach

2021 ◽  
Author(s):  
Chaojian Chen ◽  
Mikhail Kruglyakov ◽  
Alexey Kuvshinov
Keyword(s):  
Integral Equation ◽  
Three Dimensional ◽  
Region Of Interest ◽  
Electromagnetic Modeling ◽  
Multi Scale ◽  
Scale Modeling ◽  
Uniform Grids ◽  
The Matrix ◽  
Integral Equation Approach ◽  
Matrix Vector

Summary Most of the existing three-dimensional (3-D) electromagnetic (EM) modeling solvers based on the integral equation (IE) method exploit fast Fourier transform (FFT) to accelerate the matrix-vector multiplications. This in turn requires a laterally-uniform discretization of the modeling domain. However, there is often a need for multi-scale modeling and inversion, for instance, to properly account for the effects of non-uniform distant structures, and at the same time, to accurately model the effects from local anomalies. In such scenarios, the usage of laterally-uniform grids leads to excessive computational loads, both in terms of memory and time. To alleviate this problem, we developed an efficient 3-D EM modeling tool based on a multi-nested IE approach. Within this approach, the IE modeling is first performed at a large domain and on a (laterally-uniform) coarse grid, and then the results are refined in the region of interest by performing modeling at a smaller domain and on a (laterally-uniform) denser grid. At the latter stage, the modeling results obtained at the previous stage are exploited. The lateral uniformity of the grids at each stage allows us to keep using the FFT for the matrix-vector multiplications. An important novelty of the paper is a development of a “rim domain” concept which further improves the performance of the multi-nested IE approach. We verify the developed tool on both idealized and realistic 3-D conductivity models, and demonstrate its efficiency and accuracy.

Guest Column

2021 ◽  
Vol 52 (2) ◽  
pp. 46-70
Author(s):  
A. Knop ◽  
S. Lovett ◽  
S. McGuire ◽  
W. Yuan
Keyword(s):  
Machine Learning ◽  
Approximation Algorithms ◽  
Computer Science ◽  
Data Structures ◽  
Coding Theory ◽  
Communication Complexity ◽  
Property Testing ◽  
Theoretical Computer Science ◽  
Streaming Algorithms ◽  
Theoretical Computer

Communication complexity studies the amount of communication necessary to compute a function whose value depends on information distributed among several entities. Yao [Yao79] initiated the study of communication complexity more than 40 years ago, and it has since become a central eld in theoretical computer science with many applications in diverse areas such as data structures, streaming algorithms, property testing, approximation algorithms, coding theory, and machine learning. The textbooks [KN06,RY20] provide excellent overviews of the theory and its applications.

The Role of Average Health Status

2016 ◽  
pp. 884-899
Author(s):  
Jordan Panayotov
Keyword(s):  
Health Status ◽  
Population Health ◽  
Gold Standard ◽  
Health Impact ◽  
Local Context ◽  
Policy Options ◽  
The Matrix ◽  
The Right ◽  
No Gold Standard ◽  
Average Health

Economic, social and environmental policies, programs and projects have impact on health. Health in All Policies (HiAP) aims to improve population health by taking into account these impacts. HiAP needs appropriate tools for assessing impacts on population health. When making choices between policy options, decision-makers rely on predictions from Health Impact Assessment. Currently there is no gold standard for establishing and assessing validity of predictions. This paper distinguishes between two levels of causal pathways regarding health impacts – specific and conditional, and proposes the Average Health Status – Health Inequalities Matrix as gold standard. The Matrix facilitates making the right choices at any level and local context, thus is useful for researchers, policy-makers and practitioners for designing, analysing and evaluating all kinds of policies. By allowing quick, reliable and inexpensive appraisal of different policy options the matrix makes feasible taking into account the impacts on population health and paves the way for institutionalizing of HiAP.

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